A classical area within combinatorics that appears related to these questions is discrepancy theory. Let \(S_1,\ldots ,S_m \subseteq \lbrace  1,\ldots ,n\rbrace \)  be a set system over \(n\)  elements. Then
the goal is to find a bi-coloring \(\mathbf {x} \in \lbrace  -1,1\rbrace ^n\)  so that the worst imbalance \(\max _{i=1,\ldots ,m} |\sum _{j \in S_i}x_j|\)  is minimized. A seminal result of Spencer [1]} says that for \(m \ge n\) ,
the discrepancy is bounded by \(O(\sqrt{n \log (\frac{2m}{n})})\) . If no element is in more than \(t\)  sets, then
one can also prove a bound of \(2t\) , see Beck and Fiala [2]}.
A convex geometry based method by Banaszczyk [3]} shows that for any \(\mathbf {A} \in \mathbb {R}^{m \times n}\)  with column length \(\Vert \mathbf {A}^j\Vert _2 \le 1\)  for all \(j=1,\ldots ,n\)  and any symmetric convex body \(K \subseteq \mathbb {R}^m\)  with Gaussian measure at least \(1/2\)  (for example \(K = \Theta (\sqrt{\log (m)}) \cdot B_{\infty }^m\)  or \(K = \Theta (\sqrt{m}) \cdot B_2^m\)  work), there is a coloring \(\mathbf {x} \in \lbrace -1,1\rbrace ^n\) 
with \(\mathbf {A}\mathbf {x} \in 5K\) . Interestingly, neither of these cited results of [1]}, [2]}, [3]} can be obtained by merely taking a uniform random coloring \(\mathbf {x}\) .
But for example, the result by Spencer allows for elegant algorithmic proofs. While the first
such algorithm was due to Bansal [7]}, we focus on the later work of Lovett and Meka [8]} whose main claim can be paraphrased as follows:
